Finite deformation plasticity and viscoplasticity laws exhibiting nonlinear hardening rules

 This paper deals with plasticity and viscoplasticity laws exhibiting nonlinear kinematic hardening as well as nonlinear isotropic hardening rules. In Tsakmakis (1996a, b) a constitutive theory has been formulated within the framework of finite deformations, which is based on the concept of so-called dual variables and associated time derivatives. Within two families of dual variables, two different formulations have been proposed for kinematic hardening, referred to as Models 1 and 2. In particular, rigid plastic deformations without isotropic hardening have been considered. In the present paper, the constitutive theory of Tsakmakis (1996a, b) is appropriately extended to take into account isotropic hardening as well as elastic deformations. Care is taken that the evolution equations governing the hardening response fulfill the intrinsic dissipation inequality in every admissible process. For the case of small elastic strains combined with a simplification concerning kinematic hardening, to be explained in the paper, an efficient, implicit time-integration algorithm is presented. The algorithm is developed with a view to implementation in the ABAQUS Finite Element code. Also, explicit formulas for the consistent tangent modulus are derived. Received 22 September 1999     

Finite element implementation of large deformation micropolar plasticity exhibiting isotropic and kinematic hardening effects

Micropolar theories offer a possibility to model size effects in the constitutive behaviour of materials. Typical feature of such models is that they deal with a microrotation, which is supposed to represent an independent state variable, and its space gradient. As a consequence, the stress tensor is no longer symmetric and couple stresses enter the theory. Accordingly, a micropolar plasticity law exhibiting kinematic hardening effects should account for both, a back‐stress tensor and a back‐couple stress tensor. This has been considered in the micropolar plasticity model developed by Grammenoudis and Tsakmakis. The purpose of the current paper is to specify some constitutive functions in this model, to elucidate the finite element implementation as well as to demonstrate its capabilities in describing size effects. Copyright © 2005 John Wiley & Sons, Ltd.     

Eulerian constitutive model for finite deformation plasticity with anisotropic hardening

A constitutive model is presented for finite strain plasticity. The model incorporates both isotropic and kinematic hardening of the Ziegler type. The corotational rate used here is in line with the theory suggested by Paulun and Pecherski (1985) but not necessarily confined to the von Mises type yield criterion and the Prager hardening rule. The aspect of integration of the corotational rates is also discussed here. The use of the integration of the material rate of tensors with time as a substitute for the proper integration with time of corotational rates leads to mathematical inconsistencies of the theory of Lie derivatives. The problem of simple shear is investigated and compared with other works.     

Interface controlled plasticity in metals: dispersion hardening and thin film deformation

Plastic deformation and strengthening of metals, a classic subject of physical metallurgy, is still a central theme of present-day materials research. This review focusses on two modern aspects of fundamental and practical interest: the mechanism of dispersion hardening at high temperatures, which allows the design of alloys operating close to their melting point; and the constraints on dislocation and diffusional deformation processes in metallic thin films, a potential reliability problem for micro-systems subjected to high internal stresses. The commonality lies in the importance of interfacial effects: the interaction of lattice dislocations with interfaces — between particle and matrix, or between film and substrate — controls the strengthening effect in both instances; diffusional creep occurs in both cases, but is again limited by interface effects. An attempt is made to summarize the current understanding of these phenomena with special emphasis on modelling and transmission electron microscopy results.     

Mathematical and numerical model for nonlinear viscoplasticity

A macroscopic model describing elastic-plastic solids is derived in a special case of the internal specific energy taken in separable form: it is the sum of a hydrodynamic part depending only on the density and entropy, and a shear part depending on other invariants of the Finger tensor. In particular, the relaxation terms are constructed compatible with the von Mises yield criteria. In addition, Maxwell-type material behaviour is shown up: the deviatoric part of the stress tensor decays during plastic deformations. Numerical examples show the ability of this model to deal with real physical phenomena.     

Classical and Cosserat plasticity and viscoplasticity models for slow granular flow

We consider models for the rheology of dense, slowly deforming granular materials based of classical and Cosserat plasticity, and their viscoplastic extensions that account for small but finite particle inertia. We determine the scale for the viscosity by expanding the stress in a dimensionless parameter that is a measure of the particle inertia. We write the constitutive relations for classical and Cosserat plasticity in stress-explicit form. The viscoplastic extensions are made by adding a rate-dependent viscous stress to the plasticity stress. We apply the models to plane Couette flow, and show that the classical plasticity and viscoplasticity models have features that depart from experimental observations; the prediction of the Cosserat viscoplasticity model is qualitatively similar to that of Cosserat plasticity, but the viscosities modulate the thickness of the shear layer.     

Classical plasticity and viscoplasticity models reformulated: theoretical basis and numerical implementation

The well-known phenomenological model of small strain rate-independent plasticity is reformulated in this paper. The main difference from the classical expositions concerns the absence of the plastic strain from the list of state variables. We show that with the proposed choice of state variables, including the total and the elastic strains and strain-like variables which control hardening, we recover all the ingredients of the classical model from a minimum number of hypotheses: instantaneous elastic response and the principle of maximum plastic dissipation. We also show that using a regularized, penalty-like form of the principle of maximum plastic dissipation, we can recover the classical viscoplasticity model. As opposed to the previous schemes used for the finite element implementation of this model (e.g. B-bar method), we propose an approach in which the basic set of equations need not be modified. The operator split method is used to simplify the details of the numerical implementation concerning both the computation of state variables and the incompatible mode based finite element approximations. The latter proves to be indispensable for accommodating the near-incompressible deformation patterns arising in the classical plasticity. An extensive set of numerical simulations is used to illustrate the proposed formulation. © 1998 John Wiley & Sons, Ltd.     

Non-linear B-stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity

A class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward-Euler return maps as a particular case. More importantly, it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary Helmholz free energy function (B-stability). Since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm, the requirement of B-stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity. The analysis that follows depends critically on the assumption of convexity. In particular, the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation. The proposed algorithms obey the discrete counterpart of this classical principle.     

Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms

Rate-independent plasticity and viscoplasticity in which the boundary of the elastic domain is defined by an arbitrary number of yield surfaces intersecting in a non-smooth fashion are considered in detail. It is shown that the standard Kuhn-Tucker optimality conditions lead to the only computationally useful characterization of plastic loading. On the computational side, an unconditionally convergent return mapping algorithm is developed which places no restrictions (aside from convexity) on the functional forms of the yield condition, flow rule and hardening law. The proposed general purpose procedure is amenable to exact linearization leading to a closed-form expression of the so-called consistent (algorithmic) tangent moduli. For viscoplasticity, a closed-form algorithm is developed based on the rate-independent solution. The methodology is applied to structural elements in which the elastic domain possesses a non-smooth boundary. Numerical simulations are presented that illustrate the excellent performance of the algorithm.     

A finite strain model of combined viscoplasticity and rate-independent plasticity without a yield surface

A new internal variable formulation dealing with mechanisms with different characteristic times in solid materials is proposed within a finite deformation framework. The framework relies crucially on the consistent combination of a general viscoplastic theory and a rate-independent theory (generalized plasticity) which does not involve the yield surface concept as a basic ingredient. The formulation is developed initially in a material setting and then is extended to a covariant one by applying some basic elements and results from the tensor analysis on manifolds. The covariant balance of energy is systematically employed for the derivation of the mechanical state equations. It is shown that unlike the case of finite elasticity, for the proposed formulation the covariant balance of energy does not yield the Doyle–Ericksen formula, unless a further assumption is made. As an application, by considering the material (intrinsic) metric as a primary internal variable accounting for both elastic and viscoplastic (dissipative) phenomena within the body, a constitutive model is proposed. The ability of the model in simulating several patterns of the complex response of metals under quasi-static and dynamic loadings is assessed by representative numerical examples.     

Thermodynamic format and heat generation of isotropic hardening plasticity

Summary Heat generation due to plastic deformation of metals and steel is studied. Whereas in many investigations it is assumed that the fraction η of the plastic work transformed into heat is constant throughout the deformation process, the fraction η is here derived from thermodynamic considerations in a large-strain setting. It is shown that for elasto-plasticity the fraction η follows as a result of the choice of free energy, potential function and yield function. Taking the stress-strain response and the dissipative properties of the material as basis for calibration, it is shown that the thermodynamic framework of a thermoplastic material is non-unique for the general situation of non-associated plasticity. In the investigation conducted here, the mechanical response and the portion of the plastic work converted into heat (or into stored energy) during plastic deformations is predicted by means of isotropic hardening von Mises plasticity. It is shown that for a situation in which the internal variable is taken as the effective plastic, close fitting to experimental data requires a non-associated format of the evolution law for the internal variable.     

Kinematic hardening rules for modeling uniaxial and multiaxial ratcheting

Ratcheting, which is the strain accumulation observed under the unsymmetrical stress controlled loading and non-proportional loadings, is modeled using the simplified viscoplasticity theory based on overstress (VBO). The influences of kinematic hardening laws on the uniaxial and multiaxial non-proportional ratcheting behavior of CS 1026 carbon steel have been investigated. The following kinematic hardening rules have been considered: the classical kinematic hardening rule, the kinematic hardening rules introduced by Armstrong–Frederick, Burlet–Cailletaud and the modified Burlet–Cailletaud. The investigated loading conditions include uniaxial stress controlled test with non-zero mean stress, and axial strain controlled cyclic test of thin-walled tubular specimen in the presence of constant pressure. Numerical results are compared with the experimental data obtained by Hassan and Kyriakides [Hassan T, Kyriakides S. Ratcheting in cyclic plasticity, part I: uniaxial behavior. Int J Plast 1992;8:91–116] and Hassan et al. [Hassan T, Corona E, Kyriakides S. Ratcheting in cyclic plasticity, part I: multiaxial behavior. Int J Plast 1992;8:117–146]. It is observed that all investigated kinematic hardening rules do not improve ratcheting behavior under multiaxial loading, but over-prediction still exists.     

On the concept of relative and plastic spins and its implications to large deformation theories. Part II: Anisotropic hardening plasticity

Summary By resorting to both microscopic and macroscopic considerations, including the concept of single slip, dislocation stress, and a scale invariance argument we show that the notion and formalism of the relative spin introduced in Part I reduces to that of plastic spin previously recognized in the literature. The central feature of this reduction is the possibility of obtaining physically based constitutive equations for the plastic spin along with appropriate evolution equations for the dislocation or back stress. When these constitutive models are incorporated in the analysis of existing data on tension-torsion tests, we find satisfactory agreement between theory and experiment. In particular, a theoretical interpretation of the torsionally induced axial strain, as observed for example by Swift, Bailey et al., Hart and Chang, and others, is provided. Moreover, the recent experiments of Montheillet et al. on torsionally induced axial stresses are discussed in the light of the presently proposed models of large inelastic deformation inelasticity accounting for anisotropy and texture effects.

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With 12 Figures     

On the exploitation of Armstrong-Frederik type nonlinear kinematic hardening in the numerical integration and finite-element implementation of pressure dependent plasticity models

In this paper, an unconditionally stable algorithm for the numerical integration and finite-element implementation of a class of pressure dependent plasticity models with nonlinear isotropic and kinematic hardening is presented. Existing algorithms are improved in the sense that the number of equations to be solved iteratively is significantly reduced. This is achieved by exploitation of the structure of Armstrong-Frederik-type kinematic hardening laws. The consistent material tangent is derived analytically and compared to the numerically computed tangent in order to validate the implementation. The performance of the new algorithm is compared to an existing one that does not consider the possibility of reducing the number of unknowns to be iterated. The algorithm is used to implement a time and temperature dependent cast iron plasticity model, which is based on the pressure dependent Gurson model, in the finite-element program ABAQUS. The implementation is applied to compute stresses and strains in a large-scale finite-element model of a three cylinder engine block. This computation proofs the applicability of the algorithm in industrial practice that is of interest in applied sciences.     

Variation of solute distributions during deformation and bake hardening process and their effect on bake hardening phenomenon in ultra-low carbon bake hardening steels

Three-dimensional atom probe was used to investigate solute carbon and other elements distributions during bake hardening process after pre-deformation and also to analyze their effect on bake hardening phenomenon of the steels. Two different kinds of bake hardening steels were prepared and annealed by water quenching. The as-received samples were pre-deformed at different levels (from 0 to 10%), and baked at 170 °C for 20 min. Distributions and concentrations of solute elements in the steels were characterized with three-dimensional atom probe. Bake hardening values of the steels were examined by tensile experiments. Three dimensional atom probe detection results indicate that C distribution changes little with the increase of pre-deformation in BH-Mn steel. In BH-P steel, however, with the increase of pre-deformation, more C clusters form in the matrix and C concentration decreases. Distribution patterns and the maximum separation distance method results prove that the C cluster is just C segregation or C together with P segregation rather than vanadium carbides precipitate. Moreover, bake hardening experiment results indicate that BH values are similar in the two BH steels and the BH values change only a little as the pre-deformation increases from 2 to 10%. The 4% pre-deformation induces the highest BH values in the two BH steels, and is considered to be the critical pre-deformation in making the balance of Cottrell atmosphere in the two BH steels.     

Parameter identification of hardening laws for bulk metal forming using experimental and numerical approach

The simulative prediction of material behaviour in forming processes necessitates a precise determination of the material parameters. The present work focusses on the modelling of the isostatic part of the flow stress using a flow curve with an analytical suppression of the influence of friction and an adequate analytical law. The experimental data are obtained from isothermal upsetting tests with various upsetting ratios. The different ratios are based on a variation of the height of the sample, remaining the diameter constant. For the proposed flow stress law five parameters are identified. In order to decrease the number of function evaluations, a new reduction model method based on both analytical and sequential quadratic programming (SQP) algorithms is developed and applied to identify flow stress law parameters. A comparison with traditional SQP algorithm is also done. A 3D finite element model is built in order to simulate a side pressing test and an experimental validation is done. As numerical results fit very well experimental data, the proposed model achieves a precise prediction of the flow behaviour. The identification of the other parts of the model (i.e. dependencies on strain-rate and temperature) are conducted in further works.